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Mirrors > Home > MPE Home > Th. List > xmstopn | Structured version Visualization version GIF version |
Description: The topology component of an extended metric space coincides with the topology generated by the metric component. (Contributed by Mario Carneiro, 26-Aug-2015.) |
Ref | Expression |
---|---|
isms.j | ⊢ 𝐽 = (TopOpen‘𝐾) |
isms.x | ⊢ 𝑋 = (Base‘𝐾) |
isms.d | ⊢ 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋)) |
Ref | Expression |
---|---|
xmstopn | ⊢ (𝐾 ∈ ∞MetSp → 𝐽 = (MetOpen‘𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isms.j | . . 3 ⊢ 𝐽 = (TopOpen‘𝐾) | |
2 | isms.x | . . 3 ⊢ 𝑋 = (Base‘𝐾) | |
3 | isms.d | . . 3 ⊢ 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋)) | |
4 | 1, 2, 3 | isxms 23054 | . 2 ⊢ (𝐾 ∈ ∞MetSp ↔ (𝐾 ∈ TopSp ∧ 𝐽 = (MetOpen‘𝐷))) |
5 | 4 | simprbi 500 | 1 ⊢ (𝐾 ∈ ∞MetSp → 𝐽 = (MetOpen‘𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 × cxp 5517 ↾ cres 5521 ‘cfv 6324 Basecbs 16475 distcds 16566 TopOpenctopn 16687 MetOpencmopn 20081 TopSpctps 21537 ∞MetSpcxms 22924 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-rab 3115 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-xp 5525 df-res 5531 df-iota 6283 df-fv 6332 df-xms 22927 |
This theorem is referenced by: imasf1oxms 23096 ressxms 23132 prdsxmslem2 23136 tmsxpsmopn 23144 xmsusp 23176 cmetcusp1 23957 minveclem4a 24034 minveclem4 24036 qqhcn 31342 rrhcn 31348 rrexthaus 31358 dya2icoseg2 31646 sitmcl 31719 |
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